Choose your preferred view mode

Please select whether you prefer to view the MDPI pages with a view tailored for mobile displays or to view the MDPI
pages in the normal scrollable desktop version. This selection will be stored into your cookies and used automatically
in next visits. You can also change the view style at any point from the main header when using the pages with your
mobile device.

Abstract

We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For each free ultrafilter p on the set N of natural numbers and every sequence (Un) of non-empty open subsets of G, one can choose a point xn∈Un for all n∈N in such a way that the resulting sequence (xn) has a p-limit in G; that is, {n∈N:xn∈V}∈p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group G above is not pseudo-ω-bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum ⨁i∈IXi, where each space Xi is either maximal or discrete, contains no infinite separable pseudocompact subsets.
View Full-Text

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

Printed Edition Available!
A printed edition of this Special Issue is available here.